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Theory and Numerical Treatment of the Einstein Constraint Equations

Wednesday, January 25, 1PM – 2:30PM
ACE 6.304

Michael Holst

The Einstein constraints equations are of fundamental interest in mathematical physics in the study of Einstein's theory of general relativity. This coupled nonlinear elliptic system must also be solved in various forms for gravitational wave simulation. The constraint equations have been studied intensively for half a century; in this lecture we consider two closely-related problems:

(1) The development of new existence results for the constraint equations that violate the so-called "near-CMC condition";

(2) The development of "provably good" approximation methods, with a focus on adaptive finite element methods. The first problem has been open since the "conformal method" for solving the Einstein constraints was first proposed in 1973. We recently developed a new analysis approach based on combining topological fixed point arguments with the construction of "global barriers"; this framework made it possible in 2009 to establish the first existence results for the Einstein constraints without using the near-CMC condition. We will give an overview of these new results, and also outline some ongoing projects to further extend the solution theory. (Various parts of this part of the work are joint work with G. Nagy, and G. Tsogtgerel.)

Once the solution theory is in place, we can address the second problem, which splits into two clearly stated, fundamental, numerical analysis questions:

(2a) A priori error estimates for finite element-type methods for nonlinear elliptic systems posed on 2- and 3-surfaces;

(2b) Establishing proofs of contraction for adaptive finite element methods for nonlinear elliptic systems of this type. Both problems are at the heart of two extremely actively areas of research in numerical analysis; we will give just a brief overview of some of our recent relevant results in each of these areas, in an effort to produce an efficient (low complexity), robust (always works), and high-fidelity (highly accurate) numerical approximation scheme. Solving Problem (2a) will involving exploiting very recent work on finite element exterior calculus, and problem (2b) will involve extending recent work on convergence theory for linear problems to the nonlinear setting.

(Various portions of this part of the work are joint with Y. Zhu, G. Tsogtgerel, R. Bank, R. Szypowski, and S. Pollack.)

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